Learn Extracted exam questions AP Physics C: Mechanics 2026 Free Response
2026 Free Response
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A box of mass $M$ slides to the right on a horizontal surface. A small cube, also of mass $M$, is inside the box. The cube is against the right wall of the box and above the bottom of the box, as shown in Figure 1. The coefficients of static and kinetic friction between the cube and the wall of the box are $\mu_s$ and $\mu_k$, respectively, where $\mu_s > \mu_k$.
[Figure 1: Side view diagram. A box labeled "Box, $M$" slides to the right (shown by motion lines and arrows on the left). Inside the box, a small shaded cube labeled "Cube" and "$M$" rests against the inside of the right wall of the box, above the bottom of the box, labeled with $\mu_s, \mu_k$ at the contact surface. An arrow labeled $\vec{F}_R$ points left, into the box, from the right side, representing the resistive force on the box.]
The speed of the box is decreasing as a result of a resistive force that is exerted on the box. The resistive force $\vec{F}_R$ is modeled by $\vec{F}_R = -b\vec{v}$, where $b$ is a positive constant and $\vec{v}$ is the velocity of the box. Friction between the box and horizontal surface is negligible.
Consider the motion of the box and cube for the following times.
- At time $t = 0$, the box-cube system slides to the right with speed $v_0$.
- For $0 \le t < t_{\text{crit}}$, the cube remains in contact with the wall of the box at a constant height above the bottom of the box.
- At $t = t_{\text{crit}}$, the cube begins to slide downward along the wall of the box.
The magnitude of the normal force exerted on the cube by the wall of the box is $F_N$.
Determine an expression for the magnitude of the acceleration of the cube. Express your answer in terms of $M$, $F_N$, and physical constants, as appropriate.
Derive an expression for $F_N$ as a function of $t$ from $t = 0$ until immediately before the cube reaches the bottom of the box. Express your answer in terms of $M$, $b$, $v_0$, $t$, and physical constants, as appropriate. Begin your derivation with a fundamental physics principle or an equation from the reference information.
On the axes shown in Figure 2, sketch a graph of the magnitude $F_f$ of the frictional force exerted on the cube by the wall of the box as a function of $t$ from $t = 0$ until immediately before the cube reaches the bottom of the box.
[Figure 2: A blank graph with vertical axis labeled $F_f$ and horizontal axis labeled $t$, origin labeled $O$. A vertical dashed line on the horizontal axis is labeled $t_{\text{crit}}$.]
Derive an expression for $t_{\text{crit}}$ in terms of $M$, $b$, $\mu_s$, $v_0$, and physical constants, as appropriate. Begin your derivation with a fundamental physics principle or an equation from the reference information.
A projectile of total mass $4M$ is launched from the ground at position $x = 0$ and time $t = 0$. The projectile is launched with an initial speed $v_0$ at an angle $\theta$ above the horizontal. When the projectile is at the highest point in its trajectory, it breaks into Pieces Q and R of masses $M$ and $3M$, respectively.
The motion of the projectile is described for the following times.
- At $t = t_1$, immediately after the projectile breaks apart, the two pieces are moving away from each other horizontally.
- At $t = t_2$, Piece Q reaches the ground at $x = 0$ and Piece R reaches the ground at $x = x_2$, as shown in Figure 1.
[Figure 1: Three side-view snapshots labeled $t=0$, $t=t_1$, $t=t_2$, with a note "Figure not drawn to scale." At $t=0$: a $+y$ vertical axis and $+x$ horizontal axis are shown at the origin; a small box on the ground at $x=0$ launches with velocity $v_0$ at angle $\theta$ above the horizontal, following a dashed parabolic path upward. At $t=t_1$: at the top of the dashed parabolic trajectory, two pieces are shown separating horizontally — "Piece Q, $M$" moving to the left and "Piece R, $3M$" moving to the right, connected by a wavy break symbol; the ground below is marked $x=0$. At $t=t_2$: on the ground, a small box (Piece Q) sits at $x=0$ and a larger box (Piece R) sits at $x=x_2$ to the right.]
The horizontal and vertical components of a momentum vector are represented by $p_x$ and $p_y$, respectively. The shaded bars in Figure 2 represent $p_x$ and $p_y$ of the projectile immediately after $t = 0$.
[Figure 2: "Figure 2, $t=0$." A bar chart titled "Momentum" with vertical axis showing a dashed zero line labeled $0$ and horizontal categories $p_x$ and $p_y$, captioned "Complete Projectile." A shaded bar for $p_x$ rises to a medium-large positive height above the zero line; a shaded bar for $p_y$ rises to a smaller positive height above the zero line (shorter than the $p_x$ bar).]
[Figure 3: "Figure 3, $t=t_1$." Two blank bar charts titled "Momentum," each with a dashed zero line labeled $0$ and horizontal categories $p_x$ and $p_y$; the left one captioned "Piece Q" and the right one captioned "Piece R." Both are currently empty (no shaded bars drawn).]
On Figure 3, draw shaded bars to represent $p_x$ and $p_y$ of Pieces Q and R at $t = t_1$.
- Start the shaded bars at the dashed line that represents zero momentum.
- Represent the magnitudes of the components of the momentum by using the heights of the shaded bars, consistent with the scale used in Figure 2.
- Represent any momentum component that is equal to zero by drawing a distinct line on the zero-momentum line.
Derive an expression for $x_2$ in terms of $v_0$, $\theta$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
The horizontal component of a velocity vector is represented by $v_x$. Figure 4 shows the horizontal component $v_{x,\text{cm}}$ of the velocity of the center of mass of the projectile as a function of $t$ during the time interval $0 < t < t_1$.
[Figure 4: A graph with vertical axis $v_x$ and horizontal axis $t$ (origin $O$). A horizontal dashed reference line is labeled $v_{x,\text{cm}}$ above the $t$-axis, and another horizontal dashed reference line is labeled $-v_{x,\text{cm}}$ below the $t$-axis (symmetric about the axis). A solid horizontal line segment at height $v_{x,\text{cm}}$ runs from $t=0$ to $t=t_1$, representing $v_x$ during $0
On Figure 4, sketch a line or curve to represent $v_x$ as a function of $t$ for the time interval $t_1 < t < t_2$ for each of the following.
- Piece Q
- Piece R
- The center of mass of the two-piece system
Clearly label all lines or curves.
Consider a case in which the projectile is launched at the same angle and initial speed as initially described. When the projectile breaks into Pieces Q and R, Piece Q falls straight down. In this case, Piece R reaches the ground at $x = x_{\text{new}}$.
Indicate whether $x_{\text{new}}$ is greater than, less than, or equal to $x_2$ by writing one of the following.
- $x_{\text{new}} > x_2$
- $x_{\text{new}} < x_2$
- $x_{\text{new}} = x_2$
Briefly justify your answer either by referencing a feature of the representations you drew in part A or C or by using conceptual reasoning beyond algebraic solutions.
The following information applies to parts A and B.
A horizontal spring of known spring constant $k$ is attached to a wall. A block of known mass $m$ is placed on a horizontal surface next to but not attached to the spring. The spring is at its relaxed length when the block is at position $x = 0$, as shown in Figure 1. There is friction between the block and the surface only for positions $x > 0$.
[Figure 1: A horizontal diagram with a $+x$ axis arrow pointing right at the top. A spring with spring constant $k$ is attached to a wall on the left and connects to a block labeled $m$. To the right of the block (for $x>0$), the surface is marked with a squiggly friction symbol and labeled $\mu_k$. Below the block is the label $x=0$.]
A student is asked to experimentally determine the coefficient of kinetic friction $\mu_k$ between the block and the surface using a graph. The student is permitted to use only measurements from a meterstick.
Indicate quantities that could be measured by the student that would allow them to determine $\mu_k$ using a graph.
Briefly describe a method to reduce experimental uncertainty for the measured quantities.
Indicate what quantities the student could graph on the horizontal and vertical axes to create a graph that can be used to determine $\mu_k$.
Briefly describe the relationship between $\mu_k$ and a feature of the graph from part B (i). Your answer may include an equation that relates $\mu_k$ and the chosen feature of the graph.
The following information applies to parts C and D.
In a different experiment, the student is asked to determine the spring constant $k_{\text{new}}$ of a new spring that is attached to a wall. A block of mass 2.0 kg is placed next to the spring on a surface where frictional forces are negligible. The surface is horizontal near the spring and slopes upward into a ramp to the right of the initial position of the block, as shown in Figure 2.
[Figure 2: A diagram showing a spring labeled $k_{\text{new}}$ attached to a wall on the left, connected to a block labeled "2.0 kg" with an arrow "$f$" curving above it. The horizontal surface to the right of the block slopes upward into a curved ramp rising to the upper right.]
The student pushes the block, compressing the spring a distance $s$, as shown in Figure 3. The block is released from rest.
[Figure 3: A diagram showing the spring compressed, attached to the wall on the left and to the block, with a distance $s$ marked by a double-arrow between the wall-side spring end and the block's initial (uncompressed) dashed outline position. To the right, a dashed outline of the block is shown partway up the curved ramp at height $h$ above the horizontal surface level, with $h$ marked by a vertical double-arrow.]
The student uses a meterstick to measure the maximum height $h$ of the block on the ramp. The experiment is repeated several times with different compression distances $s$. The student's measurements of $s$ and $h$ are shown in Table 1.
Table 1
| $s$ (m) | $h$ (m) |
|---|---|
| 0.020 | 0.02 |
| 0.040 | 0.04 |
| 0.060 | 0.13 |
| 0.080 | 0.18 |
| 0.100 | 0.33 |
Label the axes of the grid provided with measured or calculated quantities. Include units, as appropriate. The graphed quantities should yield a linear graph that can be used to determine $k_{\text{new}}$.
On the grid provided, create a graph of the quantities indicated in part C (i).
- Clearly label the vertical and horizontal axes with numerical scales.
- Plot the corresponding data points on the grid.
- Table 2 is provided in your booklet for scratch work and will not be scored.
[Figure: Two blank labeled-axis lines reading "Quantity (units, if appropriate)" each with two blank underlines for axis labels, flanking a blank square grid for plotting, printed twice (before and after the grid) for the vertical and horizontal axes.]
Draw a best-fit line for the data graphed in part C (ii).
Using the best-fit line that you drew in part C (iii), calculate an experimental value for $k_{\text{new}}$.
A unicycle is placed on a stand such that the wheel does not touch the ground. The rotational inertia of the wheel about the axle is $I_W$. One end of a crank arm can be attached to the wheel, as shown in the side view.
[Figure: A side-view illustration of a unicycle (seat, seat post, frame, wheel with spokes, and stand legs) standing on the ground with the wheel raised off the ground, labeled "Unicycle." A dashed-line callout box magnifies the wheel region, labeled "Side View," showing the wheel's outer rim labeled "Wheel," the central hub labeled "Axle," and a straight bar extending from the axle outward labeled "Crank Arm."]
In Scenarios 1 and 2, crank arms of different lengths are connected to the wheel. In each scenario, the wheel is initially at rest. A force of magnitude $F$ is then exerted at the end of each crank arm. The direction of the force remains perpendicular to each crank arm at all times. The mass of each crank arm is negligible. The scenarios differ as described.
- Scenario 1: The length of the crank arm is $\ell_1$. The angular speed of the wheel immediately after the crank arm has been turned through one rotation is $\omega_1$.
- Scenario 2: The length of the crank arm is $\ell_2$, where $\ell_2 > \ell_1$. The angular speed of the wheel immediately after the longer crank arm has been turned through one rotation is $\omega_2$.
Indicate whether $\omega_2$ is greater than, less than, or equal to $\omega_1$ by writing one of the following.
- $\omega_2 > \omega_1$
- $\omega_2 < \omega_1$
- $\omega_2 = \omega_1$
Justify your answer. Your justification may reference equations but must include conceptual reasoning beyond algebraic solutions.
Derive an expression for the angular speed $\omega_1$ of the wheel in Scenario 1 immediately after the crank arm has been turned through one rotation. Express your answer in terms of $I_W$, $F$, $\ell_1$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
In Scenario 3, the unicycle is placed on the ground such that the unicycle can move forward without the wheel slipping on the ground. A force of the same magnitude $F$ is exerted at the end of the crank arm of length $\ell_1$, as in Scenario 1. The direction of the force remains perpendicular to the crank arm at all times. The resulting angular speed of the wheel immediately after the crank arm has been turned through one rotation is $\omega_3$.
Indicate whether $\omega_3$ is greater than, less than, or equal to $\omega_1$ by writing one of the following.
- $\omega_3 > \omega_1$
- $\omega_3 < \omega_1$
- $\omega_3 = \omega_1$
Briefly justify your answer. Your justification may reference equations but must include conceptual reasoning beyond algebraic solutions.